A. You've got some nerve, calling me at this time of night.
I would like to know if "calling" is "Gerund" or "Participle".

If "calling" is "Gerund", I wonder if “A” means 1, 2, or anything.
1. You've got a lot of nerve, in calling me at this time of night.
2. You've got a lot of nerve of calling me at this time of night.
If "calling" is "Participle", I wonder if "A" means 3 or anything..
3. You've got a lot of nerve, because you are calling me at this time of night.

Hmmmm, I suspect that you had a look at the "Fishing" thread I linked to recently. There are also many other discussions on the board that cover similar ground.

Many people have, in the past, used the term "gerund" to mean pretty well any verb + "-ing" word that is not clearly a present participle in a continuous form of the verb.

Nowadays, most grammarians restrict the usage of the term "gerund" to cases where a word is acting a a verb and a noun simultaneously, where it is doing something that only nouns to and also doing something that only verbs do. The simplest example of this is a sentence like:

"Smoking cigarettes is bad for your health." ("Smoking" has a direct object, so it must be a verb. "Smoking" is the subject of "is", so it must be a noun.) By extension, intransitive verbs that act in the same way (as subjects or objects of sentences) but which have no clearly verbal syntactical role are also referred to as gerunds:

"Smoking is bad for you."

The case you quote is somewhat different. In particular, the word "calling" is not in any way acting as a noun, so it cannot be a gerund in the strict sense. Neither can it be a deverbal noun, which acts solely as a noun, not as a verb.

In the following sentence, however, "calling" is clearly a gerund:
"Calling me this late at night is an imposition."

I have seen usages like yours labelled as "adverbial participles", but I find that deeply unsatisfying (my dislike of the use of "adverbial" to describe any construction that is difficult to classify is well documented on this site). I can see no plausible explanation why "calling" or the entire phrase "calling me this late at night" is in any way adverbial in respect of the verb "have got".

It might be possible to label "calling me this late at night" as a non-finite circumstantial clause, but that is also not entirely satisfactory, as the clause provides explanation and a reason why the speaker uttered the main clause.

Personally, I would see the clause as elliptical with a meaning something like "... as indicated by the fact that you are calling me at this time of night". Such a reading restores the full participial role of the word "calling".

It is one that grammar nerds and serious grammarians alike would argue over endlessly. Ultimately, it comes down to the school of grammar to which you belong. For a construction grammarian, the sentence is unproblematic. The entire phrase "calling me at this time of night" is a construction, as is everything else in every utterance.
For traditional grammarians, the phrase is adverbial and "calling" is an adverbial participle.

Halliwell Halliday (oops) would almost certainly have put the clause into a non-adverbial role in respect of the verb, an approach which I have been playing with in a much more radical form for several years.

The couple of posts I have written over the past weeks make me think that I should revisit my own theory of grammar that I have allowed to rest for three years or so. I use it in my everyday thinking about grammar anyway, so perhaps it is about time to tackle the many rough spots.